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Simulated Run IBM simulated run given initial price in January 2000 One year  255 trading days Drift = 5% (risk-free rate) Volatility = 0.2

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Simulated Run (contd.) IBM simulation with 3 simultaneous runs Compare with empirical data (red, solid line) Ending prices are very close Note that this run is for January

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What about predicting the future? IBM simulation for bear session for January Note how the drift rate is still positive All runs deviate from mean line and follow empirical price Ending prices are within $10 of closing price

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Accuracy? RMSD test Large vs. small values RMSD = vs for the run on the previous page

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Analysis & Conclusions Stochastic models generate price fluctuations very similar to actual data Uncertainty increases as time steps progress Further calibrations must be made to fine tune models

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Pros of Stochastic Models Inputs for stochastic models can readily be gathered from empirical data GBM model seems to fit stock price data well Risk incorporation as time increases Surprisingly accurate results Within ~$10 after one year for IBM

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Cons of Stochastic Models NO guarantee of convergence Past data plays a vital role in model performance Do stock prices always follow historical trends? There is no incorporation of current events Earnings reports Executive changes

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So, can stochastic processes predict the stock market? Unfortunately, no. Inherent unreliability Stochastic models should be only a part of the investment decision process Useful when used with traditional equity analysis Powerful tool for complex option pricing strategies